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G = D4⋊D18order 288 = 25·32

2nd semidirect product of D4 and D18 acting via D18/C18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D44D18, Q85D18, C36.51D4, C36.18C23, D36.11C22, D4⋊D96C2, C4○D43D9, C9⋊C84C22, C95(C8⋊C22), (C2×C18).9D4, (C2×D36)⋊10C2, C3.(D4⋊D6), Q82D96C2, (C3×D4).33D6, C18.60(C2×D4), (C2×C12).69D6, (C2×C4).21D18, (D4×C9)⋊4C22, (C3×Q8).57D6, (Q8×C9)⋊4C22, C4.25(C9⋊D4), C4.Dic910C2, C4.18(C22×D9), (C2×C36).46C22, C12.57(C22×S3), C22.6(C9⋊D4), C12.113(C3⋊D4), (C9×C4○D4)⋊1C2, C2.24(C2×C9⋊D4), (C3×C4○D4).12S3, (C2×C6).8(C3⋊D4), C6.108(C2×C3⋊D4), SmallGroup(288,160)

Series: Derived Chief Lower central Upper central

C1C36 — D4⋊D18
C1C3C9C18C36D36C2×D36 — D4⋊D18
C9C18C36 — D4⋊D18
C1C2C2×C4C4○D4

Generators and relations for D4⋊D18
 G = < a,b,c,d | a4=b2=c18=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 524 in 102 conjugacy classes, 38 normal (30 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C9, C12 [×2], C12, D6 [×4], C2×C6, C2×C6, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, D9 [×2], C18, C18 [×2], C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C36 [×2], C36, D18 [×4], C2×C18, C2×C18, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, C9⋊C8 [×2], D36 [×2], D36, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, C22×D9, D4⋊D6, C4.Dic9, D4⋊D9 [×2], Q82D9 [×2], C2×D36, C9×C4○D4, D4⋊D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C8⋊C22, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, D4⋊D6, C2×C9⋊D4, D4⋊D18

Smallest permutation representation of D4⋊D18
On 72 points
Generators in S72
(1 33 15 24)(2 34 16 25)(3 35 17 26)(4 36 18 27)(5 28 10 19)(6 29 11 20)(7 30 12 21)(8 31 13 22)(9 32 14 23)(37 66 46 57)(38 67 47 58)(39 68 48 59)(40 69 49 60)(41 70 50 61)(42 71 51 62)(43 72 52 63)(44 55 53 64)(45 56 54 65)
(1 63)(2 55)(3 65)(4 57)(5 67)(6 59)(7 69)(8 61)(9 71)(10 58)(11 68)(12 60)(13 70)(14 62)(15 72)(16 64)(17 56)(18 66)(19 47)(20 39)(21 49)(22 41)(23 51)(24 43)(25 53)(26 45)(27 37)(28 38)(29 48)(30 40)(31 50)(32 42)(33 52)(34 44)(35 54)(36 46)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 4)(2 3)(5 9)(6 8)(10 14)(11 13)(15 18)(16 17)(19 32)(20 31)(21 30)(22 29)(23 28)(24 36)(25 35)(26 34)(27 33)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)

G:=sub<Sym(72)| (1,33,15,24)(2,34,16,25)(3,35,17,26)(4,36,18,27)(5,28,10,19)(6,29,11,20)(7,30,12,21)(8,31,13,22)(9,32,14,23)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(41,70,50,61)(42,71,51,62)(43,72,52,63)(44,55,53,64)(45,56,54,65), (1,63)(2,55)(3,65)(4,57)(5,67)(6,59)(7,69)(8,61)(9,71)(10,58)(11,68)(12,60)(13,70)(14,62)(15,72)(16,64)(17,56)(18,66)(19,47)(20,39)(21,49)(22,41)(23,51)(24,43)(25,53)(26,45)(27,37)(28,38)(29,48)(30,40)(31,50)(32,42)(33,52)(34,44)(35,54)(36,46), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,4)(2,3)(5,9)(6,8)(10,14)(11,13)(15,18)(16,17)(19,32)(20,31)(21,30)(22,29)(23,28)(24,36)(25,35)(26,34)(27,33)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)>;

G:=Group( (1,33,15,24)(2,34,16,25)(3,35,17,26)(4,36,18,27)(5,28,10,19)(6,29,11,20)(7,30,12,21)(8,31,13,22)(9,32,14,23)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(41,70,50,61)(42,71,51,62)(43,72,52,63)(44,55,53,64)(45,56,54,65), (1,63)(2,55)(3,65)(4,57)(5,67)(6,59)(7,69)(8,61)(9,71)(10,58)(11,68)(12,60)(13,70)(14,62)(15,72)(16,64)(17,56)(18,66)(19,47)(20,39)(21,49)(22,41)(23,51)(24,43)(25,53)(26,45)(27,37)(28,38)(29,48)(30,40)(31,50)(32,42)(33,52)(34,44)(35,54)(36,46), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,4)(2,3)(5,9)(6,8)(10,14)(11,13)(15,18)(16,17)(19,32)(20,31)(21,30)(22,29)(23,28)(24,36)(25,35)(26,34)(27,33)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64) );

G=PermutationGroup([(1,33,15,24),(2,34,16,25),(3,35,17,26),(4,36,18,27),(5,28,10,19),(6,29,11,20),(7,30,12,21),(8,31,13,22),(9,32,14,23),(37,66,46,57),(38,67,47,58),(39,68,48,59),(40,69,49,60),(41,70,50,61),(42,71,51,62),(43,72,52,63),(44,55,53,64),(45,56,54,65)], [(1,63),(2,55),(3,65),(4,57),(5,67),(6,59),(7,69),(8,61),(9,71),(10,58),(11,68),(12,60),(13,70),(14,62),(15,72),(16,64),(17,56),(18,66),(19,47),(20,39),(21,49),(22,41),(23,51),(24,43),(25,53),(26,45),(27,37),(28,38),(29,48),(30,40),(31,50),(32,42),(33,52),(34,44),(35,54),(36,46)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,4),(2,3),(5,9),(6,8),(10,14),(11,13),(15,18),(16,17),(19,32),(20,31),(21,30),(22,29),(23,28),(24,36),(25,35),(26,34),(27,33),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,72),(47,71),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64)])

51 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C6A6B6C6D8A8B9A9B9C12A12B12C12D12E18A18B18C18D···18L36A···36F36G···36O
order1222223444666688999121212121218181818···1836···3636···36
size11243636222424443636222224442224···42···24···4

51 irreducible representations

dim11111122222222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D9C3⋊D4C3⋊D4D18D18D18C9⋊D4C9⋊D4C8⋊C22D4⋊D6D4⋊D18
kernelD4⋊D18C4.Dic9D4⋊D9Q82D9C2×D36C9×C4○D4C3×C4○D4C36C2×C18C2×C12C3×D4C3×Q8C4○D4C12C2×C6C2×C4D4Q8C4C22C9C3C1
# reps11221111111132233366126

Matrix representation of D4⋊D18 in GL4(𝔽73) generated by

665900
14700
4820714
53285966
,
3136431
37684246
50384237
3512365
,
287000
33100
2545453
28537042
,
454200
702800
006659
00667
G:=sub<GL(4,GF(73))| [66,14,48,53,59,7,20,28,0,0,7,59,0,0,14,66],[31,37,50,35,36,68,38,12,4,42,42,36,31,46,37,5],[28,3,25,28,70,31,45,53,0,0,45,70,0,0,3,42],[45,70,0,0,42,28,0,0,0,0,66,66,0,0,59,7] >;

D4⋊D18 in GAP, Magma, Sage, TeX

D_4\rtimes D_{18}
% in TeX

G:=Group("D4:D18");
// GroupNames label

G:=SmallGroup(288,160);
// by ID

G=gap.SmallGroup(288,160);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,675,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^18=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

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